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Leibniz

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1. Along the way toward expressing the thoughts of God before the creation of the world, Hegel’s logic consumes the possibility of mathematics at the highest moment in the doctrine of being. Just before his explicit treatment of quantity, however, he includes a note on Leibniz’ monadology and observes that “plurality remains as a fixed fundamental determination, so that the connection between [monads] falls only in the monad of monads, or in the philosopher who contemplates them”. What Hegel has grasped only vaguely here is that for Leibniz mathematics and metaphysics express the same thought, i.e., that mathematics understands the world in the same way as the divine intellect (which is the real meaning of his remark at the determination of a maximum is the work of the divine mathematician who determines the greatest number of compossibles in a given world). Leibniz’ “new mathematics”, he says, “makes man commensurate with God”.

The problem of plurality to which Hegel refers is Leibniz’ notion that the infinite (number of) monads are representations of a single universe (Monadology §78) without thereby understanding this universe as substance.* Leibniz struggles to provide an adequate topological model of such a universe** and instead speaks of the “accommodation” or harmony of all things.

*One is tempted to say “Spinozist” substance were Spinoza’s definition of substance as “one” not problematic from a mathematical point of view and which would require extensive work in disambiguation. Rather, we might safely say here “Aristotelian” substance up to and including Heidegger’s interpretation of ousia.

**Elsewhere I have claimed that such a model would be something like a Klein bottle.

2. Yet we should remember that the essence of harmony is a fundamental gap or discontinuity in what the sensibility desires as unity. The law of the series that guarantees the immanence of the world in the monad (what Badiou calls the “absolute interiority” of the monad) allows us to speak of the monad as one in a strictly different sense than that of the universe.

Here we might benefit from recalling that this is the Platonic problem par excellence. Against the Aristotelian dictum that being is always a being (i.e., that unity follows immediately from being)—and Aristotle’s well-known confusion of the Indefinite Dyad as two “counted-as-one”—Plotinus’ account of substantial number accounts both for the ontogenetic differentiation of being (see, e.g., Enneads VI.6.15) and for the fact that the One is not enumerable. What is at stake, philosophically if not mathematically, in Platonist mathematics is precisely the capacity to distinguish the one in the order of intelligibility from the unity of any individual. Being, for Plotinus, exists only because it inherits unified number from the One and, conversely, multiplicity is not the division of the One but the intellect’s contemplation of the One. We might say that substantial number is the “form” of the monad—as the immediate image of the One—combined with the “matter” of the Indefinite Dyad or, in perhaps more precise language, the Indefinite Dyad is nothing other than the limitation of unity as apostasis (and reciprocally, according to the Neopythagorean conception of monadic number, the monad is the limit of quantity: the monadic number is a progression to and a regression from mulitiplicity), the intellect is nothing other than substantial number, which is why being is not itself number but number is the principle of being.

3. What does it mean, then, to be a thinker of the One? Or, perhaps more modestly, what is at stake is the character of our ethics. For a thinker of the One, ethics is beyond being, in a sort of pagan transcendence of that which cannot be counted-as-one, as opposed to an ethics of the void, which must resist, perhaps violently, the capacity for being named and that must tear itself away from the very conditions of its survival. Our choice, however, is not that between excess and subtraction since the Plotinian One is nothing other than a series of negations: not to move away and not to progress “even a little” to the two. If there is not a symmetry between these two orientations, our choice seems to be in what direction this negation operates: whether the difference that counts is a negation of the given (multiplicity) or in the (im)possibility of negating what does not exist (a double negation!).

1. In a fairly late text (Tentamen Anagogicum, 1696), Leibniz declares that there are two “kingdoms” in nature that “interpenetrate without confusing or interfering with each other”: power and wisdom. The former denotes the “interior” relations of forces and efficient causes (i.e., physics) while the latter denotes the architectonic domain of final causes (the totality of formal determinations), i.e., metaphysics. The doctrine of pre-established harmony has the consequence that the reality of possibility is simply thought itself (Mercer reminds us that the doctrine of pre-established harmony is an extension of the sympathetic participation of each individual with the divine essence). Only a small but important difference separate Leibniz and Berkeley here: for Leibniz, ideas are not real beings but collapsing the distinction between ideas and spirits simply radicalizes Leibniz’s immanentism: individuals do not “have” or “contain” ideas but simply are ideas. An intentional idea is at the same time a reflexive idea just as the productive understanding of God is self-understanding.

But because Leibniz refuses the absolute immanence to which his doctrine is compelled, he must explain the difference between the confused and highly mediated understanding of the individual from that of God. And it is here that he introduces the notion of a “point of view”: the monad simply is a point of view. But instead of the optics so important for Descartes and Berkeley, Leibniz gives us topology (analysis situs). Berkeley’s optics raises space to the status of a third thing between perceiver/d; Leibniz’s conception of geometry provides us with the formal analysis of form as an account of perception in the monads (qua phenomenal) that, at the same time, explains their irreducible multiplicity (qua ontological): “the theory of similarities or of forms lies beyond mathematics and must be sought in metaphysics” (“On Analysis Situs”).

Yet perhaps the place Leibniz reserved for transcendence is merely a sign whose mode of signification is exactly how he would describe its referent: i.e., whose center is everywhere and circumference nowhere. In a letter to Mason, Leibniz affirms the principle that “a living being cannot die unless the whole universe dies (or perishes) as well”. But the theodical doctrine should not be read merely as a moral principle: this world—as the “optimal” possible world—is thus necessary. Yet it is Leibniz and not Spinoza who is ridiculed for his commitment to the necessity of the world when both men are committed to the doctrine of universal determinism as the ultimate the condition of possibility for the intelligibility of the world. If there is transcendence in Leibniz, it must consist in the possibility of rigorously differentiating the monad he calls God from any other monad.

2. The idealist tradition after Kant recognized that his decisive maneuver in the critical turn was to provide an account of transcendence only on the basis and possibility of immanence, i.e., that the restrictions on the use of concepts are legislated by reason itself: the immanence of thought and the transcendence of world (which goes under the quasi-religious name of finitude). Every idealist—and, for that matter, materialist—after Kant has repeated and re-affirmed this observation: that transcendence (to be beyond being, for example) must be immanent to itself and that immanence, being “in-itself”, must transcend itself (else it is not “in-itself”). To escape this dialectical solution, Deleuze proposes to conceive of immanence not as being-in-itself but as difference. Yet at the same time he declares the univocity of being as a redress to the Kantian legacy: philosophy concerns not the thought of difference (Hegel, Heidegger), which in any case must lead to idealism. Philosophy itself is nothing other than the expression of difference; the “method” of philosophy is deterritorialization or virtualization. This is why the plane of immanence is “the image of thought, the image thought gives itself of what it means to think” (and why the domain of the concept is the virtual while, importantly, the “discursive power of the function” pertains to the actual). The plane of immanence—as the critical point between the virtual and the actual—names the volatile unity of thought and being. Philosophy—this movement toward the virtual—would not be possible if the virtual were merely thought as “potential” or, similarly, if becoming were thought as the movement from the virtual (to the actual). The attempt to identify being and the event destroys philosophy. The plane of immanence names the condition of possibility for philosophy by the non-reversability of the lines from the virtual to the actual and in the other direction—because we must pass through the event. This and nothing more is meant by “materialism”: the plane of immanence not as a substratum but as the duality of thought and nature, neither in-itself (being) nor for-itself (act, entelechy), but affection or life.

“I’ve been wondering if it does not immediately sound crazy to you to recouch the distinction between the virtual and the actual in Difference and Repetition less as about two ontological realms (hence running the risk of something like Badiou’s criticisms of equivocity) and more as a noumenal/ontological and phenomenal/epistemic distinction in line with a kind of Leibnizian epistemology. In other words, why does Kant have to be either an epistemologist or an ontologist? Leibniz’s mistake according to Kant was, as far as I can tell, thinking he could rely on an ontological realm of monads that provided the sufficient reasons for the various perceptions expressed by monads in the activity of perception. On this account, however, we get the genesis of perceptible objects, which are also merely counted as one. What phenomenally/epistemically appears to us is conferred unity via the perceptual activity of the monad when in reality, that phenomenal unity (a real phenomenon) is built up out of an infinity of minute “petite” perceptions (it would, of course, be dogmatic to assume such silliness for Kant).

“If, as some want to do, we want to say that the virtual is the sufficient reason of the actual and all determination proceeds from it to actualization, why not make some similar move in Deleuze? I.e., at an ontological level, things are clamorous, yet at the representational/perceptual/actual level, things are synthesized/counted as one.”

The question here seems to concern the possibility of accepting the gambit of transcendental philosophy that forces a decision on how to mediate the unity of thought and being. The transcendental illusion is the failure to recognize that the domain of immanence is the use of concepts, which is what separates dogmatic metaphysics from critical philosophy. But what is the latter’s metaphysics?

Kant’s famous argument that “existence is not a predicate” is revisited in the KRV not simply as a modal principle: the “possibility that nothing exists” is self-contradictory insofar as such a possibility destroys the very notion of possibility. Hence, Kant says, “all concepts of negations are … derivative, and the realities are what contain the data and, so to speak, the matter or the transcendental content for the possibility and thoroughgoing determination of all things” (A575/B603) and says that this determination is a “transcendental substratum in our reason” which is “nothing other than the idea of a total reality (omnitudo realitatis)” (A576/B604).

But to arrive at metaphysics, we must go further. In the Opus Postumum, Kant says that God is “the most perfect in respect of every purely thought quality (ens summum, summa intelligentia, summum bonum). All these concepts are united in the distinctive judgment: God and the world—in the real division of the negative or contrarie oppositum, which the totality of being comprehends. Both are a maximum … the one as object of pure reason, the other as sense-object. Both are infinite: the first as magnitude of appearance in space and time; the second according to degree (virtualiter), as limitless activity with regard to forces (mathematical or dynamic magnitude of sense-objects)”. Kant does not retreat from the doctrine of God as a regulative ideal into the dogmatic, speculative path that begins with the unconditioned and proceeds, a priori, through the entire series of the world to arrive at the contingent individual. Rather, for both Kant and Leibniz the function God as a structural principle, more than the metaphysical principle of the ens realissimum, promises the unity of a world (of experience). Leibniz not only refuses the identity of God and substance—in the name of infinitely many substances—but preserves a single place for transcendence.

But do we really have absolute transcendence? God and world or God or world? What is Leibniz’s world? Monads are not in a world: the world does not exist outside or apart from the monads. Perception is not of a world but perception is the world obscurely and incompletely expressed by each monad (hence there is no distinction between metaphysics and epistemology for Leibniz). But exactly the same is true for God: hence the doctrine of compossibility arises from the identity of perception and understanding in God. Kant’s critical turn simply inverts the Leibnizian schema insofar as, for Leibniz, perception precedes and conditions understanding (hence the limits of our understanding is one of degree and not kind with respect to that of God’s—Monadology §60). In Kant’s terms, Leibniz’s dogmatism consists in the fact that there is nothing other than phenomena. In this (local) sense, there is only immanence.

Leibniz derives the infinity of individual monads from this immanence in the doctrine of compossibility: the individual is composed of singularities and a world consists of the convergence of singularities. The question of “real possibility” is retained in Kant as a problematic (and hence dialectical) notion. But is not virtuality nothing other than a real possibility? The greatest mistake of immanentism has been to confuse virtuality with (abstract) possibility or Aristotelian potential. Deleuze is explicit on this point in Difference and Repetition: “the virtual is opposed not to the real but to the actual. The virtual is fully real … Exactly what Proust said of states of resonance must be said of the virtual: ‘Real without being actual, ideal without being abstract’ …” But the distinction between the virtual and the actual is not a numerical difference (such would violate the univocity of being). The concept here is Bergsonian: the virtual, memory, or the past is what is most fully real. There are two possible ways, then, to describe the actual: either as a subtraction from or contraction of the virtual (e.g., the famous cone of memory in Matter and Memory) or as the folding of the virtual, that is to say, a self-limitation of the virtual (that is experienced as tendency, futurity, or time).

It is precisely this immanence lurking in the KRV that Fichte and Maimon exploited: the reality of transcendental apperception that threatens to collapse the division Kant proposes between thought and being. Leibniz, Fichte, and Maimon converge in Deleuze, perhaps, as well on this point: that thought occurs in an “intensive space” and that, consequently, metaphysics provides a genetic account of being whereas physics is the account of the actual.